### New Gauge Symmetries from String Theory

Pei-Ming Ho

Physics Department

National Taiwan University

Sep. 23, 2011@CQSE

### Gauge vs Global

For covariant quantities

Old definitions in some textbooks:

*U = independent of spacetime global*
*U = function of spacetime gauge (local)*

Gauge vs Global (2)

• Translation symmetry is usually considered as a global symmetry.

• Different interpretation of the transformation:

*Translation by a specific length L can be *

“gauged” (defined as a gauge symmetry)

equivalence:

(compactification on a circle).

• Gauge potential is useful but not necessary.

*x(t) » x(t) + L*

Gauge vs Global (3)

• 2nd example:

**base space = R**_{+}** with Neumann B.C. at x = 0.**

• 3rd example:

space an interval of length L/2 w.

Neumann B.C.

Space/(subgroups of isometry) orbifolds

*x® - x,* f(*x) ®* f(-x)

Gauge vs Global (4)

New (better) definition:

• Gauge Symmetry

– Transformation does not change physical states.

– A physical state has multiple descriptions.

– Transformation changes descriptions.

• Global Symmetry

– Transformation changes physical states.

This is a more fundamental distinction than spacetime-dependence.

### Non-Abelian Gauge Symmetry

### Modification/Generalization?

### d *A*

^{(1)}

*= dL*

^{(0)}

*+[A*

^{(1)}

### , L

^{(0)}

### ] *F*

^{(2)}

*= dA*

^{(1)}

*+ A*

^{(1)}

*A*

^{(1)}

### Þ d *F*

^{(2)}

*= [F*

^{(2)}

### , L

^{(0)}

### ]

### Non-Commutative Gauge Theory

• D-brane world-volume theory in B-field background. [Chu-Ho 99, Seiberg-Witten 99]

• Commutation relation determined by B-field.

• U(1) gauge theory is non-Abelian.

[*x** ^{i}*,

*x*

*]*

^{j}*= iQ*

^{ij}### Lie 3-Algebra Gauge Symmetry

• BLG model for multiple M2-branes ^{[07]}.

• needed for manifest compatibility with SUSY.

(ABJM model does not have manifest full SUSY.)

[T* ^{a}*,

*T*

*,*

^{b}*T*

*]*

^{c}*= f*

^{abc}*d*

*T*

^{d}*A = A*

*(*

_{iab}*T*

^{a}*Ä T*

*)*

^{b}*dx*

^{i}(*D** _{i}*f)

*= ¶*

_{a}*f*

_{i}

_{a}*+ A*

*f*

_{ibc}

_{d}*f*

^{bcd}*a*

### Generalization of Lie 3-algebra

*• All Lie n-algebra gauge symmetries are special *
cases of ordinary (Lie algebra) non-Abelian

gauge symmetries.

[T^{a}^{1} , *T*^{a}^{2} *,, T*^{a}* ^{n}*]

*= f*

^{a}^{1}

^{a}^{2}

^{a}

^{n}

^{a}

_{n+1}*T*

^{a}

^{n+1}*A = A*

_{a}_{1}

_{a}_{2}

_{a}*(*

_{n-1}*T*

^{a}^{1}

*Ä T*

^{a}^{2}

*ÄÄ T*

^{a}*) f = f*

^{n-1}

_{a}*T*

^{a}[*A,* f] *= A*_{a}_{1}_{a}_{2}_{a}* _{n-1}*f

_{a}*n* *f* ^{a}^{1}^{a}^{2}^{a}^{n}^{a}_{n+1}*T*^{a}^{n+1}

### More Gauge Theories

• Abelian Higher-form gauge theories

• Self-dual gauge theories

• Non-Abelian higher-form gauge theories

• NA SD HF GT w. Lie 3 (on NC space?)

*A*^{(}* ^{n)}* =

_{n!}^{1}

*A*

_{i}_{1}

_{i}_{2}

_{i}

_{n}*dx*

^{i}^{1}

*Ùdx*

^{i}^{2}

*ÙÙdx*

^{i}

^{n}*F*

^{(}

^{n+1)}*= dA*

^{(}

^{n)}d*A*^{(n)} *= dL*^{(n-1)}
d*F*^{(n+1)} = 0

### Dirac Monopole

• Jacobi identity (associativity)

is violated in the presence of magnetic monopoles.

• Distribution of magnetic monopoles gauge bundle does not exist

2-form gauge potential (3-form field strength) (Abelian bundle Abelian gerbe)

Q: How to generalize to non-Abelian gauge theory?

[[D* _{i}*,

*D*

*],*

_{j}*D*

*]*

_{k}*+[[D*

*,*

_{j}*D*

*],*

_{k}*D*

*]*

_{i}*+[[D*

*,*

_{k}*D*

*],*

_{i}*D*

*] = 0*

_{j}### Self-Dual Gauge Fields

• Examples:

type IIB supergravity type IIB superstring M5-brane theory twistor theory

• D=4k+2 (4k) for Minkowski (Euclidean) spacetime.

### Self-Dual Gauge Fields

*• When D = 2n, the self-duality condition is*

• a.k.a. chiral gauge bosons.

• How to produce 1^{st} order diff. eq. from action?

Trick: introduce additional gauge symmetry

*F = *F*

*F =* _{n!}^{1} *F*_{i}_{1}_{i}_{2}_{i}_{n}*dx*^{i}^{1} *Ù dx*^{i}^{2} *ÙÙ dx*^{i}^{n}

*F = _{n!}^{1} _{(}_{D-n)!}^{1} e _{i1i2iD}*F* ^{i1i2in}*dx*^{i}^{n+1}*Ù dx*^{i}^{n+2}*ÙÙ dx*^{i}^{D}

### Chiral Boson in 2D

[Floreanini-Jackiw ’87]

• Self-duality

• Lagrangian

• Gauge symmetry

• Euler-Lagrange eq.

• Gauge transformation

• 1-1 map btw space of sol’s and chiral config’s

### f ˙ _{- ¢ } f = 0

*L =* ^{1}_{2} f ¢ ( ˙ f _{- ¢ }f )

¶* _{x}*(¶

*-¶*

_{t}*)f = 0*

_{x}

Þ (¶* _{t}* -¶

*)f*

_{x}*= g(t)*

• Due to gauge symmetry, f is not an observable.

• k=f*’ * is an observable the Lagrangian is

• Formal nonlocality is unavoidable, although physics is local.

• Equivalent to a Weyl spinor in 2D in terms of the density of solitons.

• Lorentz symmetry is hidden.

*L =* ^{1}_{2} k(*t, x)*

## ( ò

*dy*e(

*x- y) ˙*k (

*t, y)*

^{-}

^{k}

^{(t, x)}

## )

[

### f

(*t, x),*

### f

(*t, y)] = i*

### e

(*x- y)*

### Comments

• No vector potential needed for gauge symmetry Vector potential is useful for defining

cov. quantities from deriv. of cov. quantities.

But it is not absolutely necessary.

• The formulation can be generalized to
self-dual higher-form gauge theories.^{ }

*D*_{m} = ¶_{m} *+ A*_{m}

*Example: M5-brane theory (D=5+1) *

[Howe-Sezgin 97, Pasti-Sorokin-Tonin 97, Aganagic-Part- Popescu-Schwarz 97, …]

[Chen-Ho 10]

[Ho-Imamura-Matsuo 08]

### Questions

• What is the geometry of Abelian higher-form gauge symmetry?

(bundles gerbes?)

• How to define non-Abelian higher-form gauge symmetry?

• Can we generalize the notion of covariant derivative and field strength?

• Do we still need the covariant derivative?

### Non-Abelian Self-Dual 2-Form Gauge Potential

• M5-brane in the (3-form) C-field background.

[Ho-Matsuo 08, Ho-Imamura-Matsuo-Shiba 08]

• Non-Abelian gauge symmetry for 2-form gauge potential

• Nambu-Poisson algebra (ex. of Lie 3-algebra) (Volume-Preserving Diffeomorphism)

• Part of the gauge potential VPD

• 1 gauge potential for 2 gauge symmetries!

[Ho-Yeh 11]

### Non-Local Non-Abelian

### Self-Dual 2-Form Gauge Field

• Need non-locality to circumvent no-go thms for multiple M5-branes. [Ho-Huang-Matsuo 11]

• In “5+1” formulation, decompose all fields

into zero modes vs. non-zero modes in the 5-th dimension.

• Zero modes 1-form potential A in 5D

• Non-zero modes 2-form potential B in 5D

### Comments

• Self-duality: Instantons, Penrose’s twistor theory applied to Maxwell and GR.

• In 4D, 2-form ≈ 0-form, 3-form ≈ (-1)-form, 4-form ≈ trivial through EM duality.

• Expect more new gauge symmetries from string theory to be discovered.

• “Symmetry dictates interaction” -- C. N. Yang.